A066339 Number of primes p of the form 4m+1 with p <= n.

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11

Offset: 1

Author

Sharon Sela (sharonsela(AT)hotmail.com), Jan 01 2002

Keywords

• nonn

Comments

Asymptotic expansion: a(n) ~ pi(n)/2 ~ n/(2log(n)) (pi(n) is in sequence A000720).

Partial sums of A079260. - Reinhard Zumkeller, Feb 06 2014

Links

• T. D. Noe, Table of n, a(n) for n=1..10000
• R. Breusch, An Asymptotic Formula for Primes Of The Form 4n+1

Crossrefs

Cf. A000720, A066490, A296021.

Programs

Haskell

a066339 n = a066339_list !! (n-1)
a066339_list = scanl1 (+) $ map a079260 [1..]
-- Reinhard Zumkeller, Feb 06 2014

Mathematica

Table[ Length[ Select[ Union[ Table[ Prime[ PrimePi[i]], {i, 2, n}]], Mod[ #, 4] == 1 & ]], {n, 2, 100} ]

PARI

for(n=1,200,print1(sum(i=1,n,if((i*isprime(i)-1)%4,0,1)),","))

Formula

a(n) + A066490(n) = A000720(n) - 1 for n >= 2. - Jianing Song, Apr 28 2021

Extensions

More terms from Robert G. Wilson v, Jan 03 2002